A372162 E.g.f. A(x) satisfies A(x) = exp( x * sqrt(1+2*x) * A(x) ).
1, 1, 5, 31, 329, 4201, 70357, 1374703, 31888817, 839198737, 25021698821, 827967913279, 30240609486265, 1205630521463161, 52177446181578005, 2434309587346377871, 121857094322821338593, 6513265883385904609057, 370302655720337288548741
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Keywords
Links
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*sqrt(1+2*x))))) -
PARI
a(n) = n!*sum(k=0, n, (k+1)^(k-1)*2^(n-k)*binomial(k/2, n-k)/k!);
Formula
E.g.f.: A(x) = exp( -LambertW(-x * sqrt(1+2*x)) ).
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * 2^(n-k) * binomial(k/2,n-k)/k!.
From Vaclav Kotesovec, Apr 21 2024: (Start)
E.g.f.: -LambertW(-x*sqrt(1 + 2*x))/(x*sqrt(1 + 2*x)).
a(n) ~ sqrt(3*r + 1) * n^(n-1) / ((1 + 2*r)^(3/4) * exp(n - 1/2) * r^(n + 1/2)), where r = (exp(2/3) + (-exp(1) + (6*(9 + sqrt(81 - 3*exp(2))))/exp(1))^(2/3)) / (6*(54 - exp(2) + 6*sqrt(81 - 3*exp(2)))^(1/3)) - 1/6 = 0.292252770550601628... (End)